1. a) The tenant should match the outside offer if the offer is less or equal her own valuation of the house
b) If the outside buyer values the house at v she would expect the tenant's value to be (v+100)/2. So, for v=120, b=110
c) The seller expected revenue is the second highest bid, which for a uniform distribution [100,160] is 100 + (160-100)/3 = 120
2. b2=(v2)^1/2
P(win) = P(b2<b1) = P((v2)^1/2<b1) = P(v2<b1^2) = b1^2
E(G1) = (v1 - b1) (b1^2) = b1^2.v1-b1^3
d.E(G1)/d.b1 = 2b1.v1 - 3b1^2 = 0 => b1=2/3 v1
If the expected bid b2 is (½ v1)^½ when v1 is the loweest bidder
In the uniform distribution [0,1] the highest expected valuation is 2/3 and the lowest is 1/3
If bidder 1 is the higher valuator, then b2=(1/3)^(1/2) = 0.577 and b1 is (as above) (2/3)(2/3) = 0.444. Turns out that the second bidder will win the auction even though he is not the one who values it the most.
3. vi=[0,15]
Expected gain is the probability of winning times the difference between my valuation and my bid minus the entrance fee:
a) p(b1>b2) = b1/15
E(G1) = b1/15 (v1 - b1) - e
d.E(G1)/d.b1 = 0 => b1 = v1/2
Therefore E(G1) = v1/30 (v1/2) - e = v1^2/60 - 2.5 > 0 => v1>12.25
So I'll enter the auction if my valuation is higher than 12.25
b) p(b1>b2>b3) = (b1/15)^2
E(G1) = (b1/15)^2 (v1 - b1) - e
d.E(G1)/d.b1 = 2.v1.b1/15 - 3 b1^2/15 = 0 => b1=2/3 v1
Therefore E(G1) = (2/45 v1)^2 (1/3 v1) - 2.5 = 0 => v1 > 15.6
So I'll never enter the auction
The above conclusions apply for any auction format as long as the bidders are risk averse
4. a and b are false as the expected revenue for the seller is always the second highest bidder's valuation, as long as the sellers are risk averse
5. The expected valuations are 0.2, 0.4, 0.6 and 0.8 therefore for both auctions the expected price is 0.6
6. a.The probability of Robert winning is 1, so he'll pay whatever Steven values for the good. For an uniform distribution [0,10], the value is 5
b. The expected revenue for the seller with a minimum bid level of 10 will be 10, as Robert will certainly bid for the object. For a minimum bid level b, the expected revenue is:
E(R) = p(v>b).b
Chapter 5 - Auctions
1. a) The tenant should match the outside offer if the offer is less or equal her own valuation of the house
b) If the outside buyer values the house at v she would expect the tenant's value to be (v+100)/2. So, for v=120, b=110
c) The seller expected revenue is the second highest bid, which for a uniform distribution [100,160] is 100 + (160-100)/3 = 120
2. b2=(v2)^1/2
P(win) = P(b2<b1) = P((v2)^1/2<b1) = P(v2<b1^2) = b1^2
E(G1) = (v1 - b1) (b1^2) = b1^2.v1-b1^3
d.E(G1)/d.b1 = 2b1.v1 - 3b1^2 = 0 => b1=2/3 v1
If the expected bid b2 is (½ v1)^½ when v1 is the loweest bidder
In the uniform distribution [0,1] the highest expected valuation is 2/3 and the lowest is 1/3
If bidder 1 is the higher valuator, then b2=(1/3)^(1/2) = 0.577 and b1 is (as above) (2/3)(2/3) = 0.444. Turns out that the second bidder will win the auction even though he is not the one who values it the most.
3. vi=[0,15]
Expected gain is the probability of winning times the difference between my valuation and my bid minus the entrance fee:
a) p(b1>b2) = b1/15
E(G1) = b1/15 (v1 - b1) - e
d.E(G1)/d.b1 = 0 => b1 = v1/2
Therefore E(G1) = v1/30 (v1/2) - e = v1^2/60 - 2.5 > 0 => v1>12.25
So I'll enter the auction if my valuation is higher than 12.25
b) p(b1>b2>b3) = (b1/15)^2
E(G1) = (b1/15)^2 (v1 - b1) - e
d.E(G1)/d.b1 = 2.v1.b1/15 - 3 b1^2/15 = 0 => b1=2/3 v1
Therefore E(G1) = (2/45 v1)^2 (1/3 v1) - 2.5 = 0 => v1 > 15.6
So I'll never enter the auction
The above conclusions apply for any auction format as long as the bidders are risk averse
4. a and b are false as the expected revenue for the seller is always the second highest bidder's valuation, as long as the sellers are risk averse
5. The expected valuations are 0.2, 0.4, 0.6 and 0.8 therefore for both auctions the expected price is 0.6
6. a.The probability of Robert winning is 1, so he'll pay whatever Steven values for the good. For an uniform distribution [0,10], the value is 5
b. The expected revenue for the seller with a minimum bid level of 10 will be 10, as Robert will certainly bid for the object. For a minimum bid level b, the expected revenue is:
E(R) = p(v>b).b
(30-b)/20 . b
(-b^2 + 30 b)/20Therefore for b
15, E(R)
11.25And getting d.E(R)/d.b
(-2b + 30)/20
0 => b = 15 is the optimum minimum bid level